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That's wrong.
There are two solutions for 4n² - 2n - 1 = 0
They just aren't integers, that's why you didn't find them.

Hahaha ... It gets interesting ... My mathematical skills got rusty. I didn't even think about the quadratic formula, last time used it in mid 90's, I think.

Two solutions are (1+sq.rt(5))/4 & (1-sq.rt(5))/4 where, a = 4, b = -2 & c = -1